[time 659] Cylindrical But Locally Lorentzian Universes

Stephen P. King (stephenk1@home.com)
Thu, 02 Sep 1999 22:56:09 -0400

Hi All,

        My friend Paul Hanna pointed me to this page. It is very similar to
what he is working on!



Cylindrical But Locally Lorentzian Universes

Cylindrical But Locally Lorentzian Universes

A three-dimensional space that is everywhere locally Eulcidean and 
yet cylindrical in all directions can be constructed by embedding the 
three spatial dimensions in a six-dimensional space according to the 

       x1 = R1 cos x/R1       x2 = R1 sin x/R1
       x3 = R2 cos y/R2       x4 = R2 sin y/R2
       x5 = R3 cos z/R3       x6 = R3 sin z/R3

so the spatial Euclidean line element is

 dx1^2 + dx2^2 + dx3^2 + dx4^2 + dx5^2 + dx6^2 = dx^2 + dy^2 + dz^2

giving a Euclidean spatial metric in a closed three-space with total 
volume (2*pi)^3*R1*R2*R3.  It's been suggested (by Klaus Kassner,
among others) that we can subtract this from an ordinary temporal 
component to give an everywhere-locally-Lorentzian spacetime that
is cylindrical in the three spatial directions, i.e.,

        ds^2 = c^2 dt^2 - (dx^2 + dy^2 + dz^2)

However, this "subtraction" doesn't seem well motivated.  One way 
of providing a motivation, and in the process making the universe 
cylindrical in ALL directions, temporal as well as spatial, would 
be to embed the entire 4D spacetime into a space of 8 dimensions, 
2 of which are purely imaginary, like this

     x1 =   R1 cos x/R1       x2 =   R1 sin x/R1
     x3 =   R2 cos y/R2       x4 =   R2 sin y/R2
     x5 =   R3 cos z/R3       x6 =   R3 sin z/R3
     x7 = i R4 cos t/R4       x8 = i R4 sin t/R4

leading (again) to a locally Lorentzian 4D metric

    (ds)^2  =  (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2

but now *all four* of the dimensions x,y,z,t are periodic.  So
here we have an everywhere-locally-Lorentzian manifold that is
closed and unbounded in every spatial and temporal direction.
(Obviously this manifold contains closed time-like worldlines.)

This reminds me a bit of Stephen Hawking's recent attempts to
describe a universe that is finite but unbounded in time as well
as space.  He too invokes an imaginary time dimension, but aside
from that I don't think his model is related to the locally-flat
cosmology described here.  Anyway, the causal structure of such
a universe is interesting.

We might imagine that a flat, closed, unbounded universe of this 
type would tend to collapse if it contained any matter, unless a 
non-zero cosmological constant is assumed.  On the other hand, I'm 
not sure what "collapse" would mean in this context.  It might 
mean that the R parameters would shrink, but R is not a dynamical
parameter of the model.  The 4D field equations operate only on
x,y,z,t.  Also, any "change" in R would imply some meta-time
parameter T, so that all the R coefficients in the embedding
formulas would actually be functions R(T).

It seems that the flatness of the 4-space is independent of the
value of R(T), and if the field equations are satisfied for one
value of R they would be satisfied for any value of R.  But I'm
not sure how the meta-time T would relate to the internal time
t for a given observer.  It might require some "meta field
equations" to relate T to the internal parameters x,y,z,t.
Possibly these meta-equations would allow (require?) the value
of R to be "increasing" versus T, and therefore indirectly
versus our internal time t = f(T), in order to achieve stability.

On the general method of embedding a locally flat n-dimensional
space in a flat 2n-dimensional space, I wonder if every orthogonal
basis in the n-space maps to an orthogonal basis in the 2n-space
according to a set of formulas formally the same as those shown
above.  If not, is there a more general mapping that applies to
all bases?

By the way, the above totally-cylindrical spacetime has a natural
expression in terms of "octonion space", i.e., the Cayley algebra 
whose elements are two ordered quaterions

     x1 = i R1 cos x/R1       x2 = i R1 sin x/R1
     x3 = j R2 cos y/R2       x4 = j R2 sin y/R2
     x5 = k R3 cos z/R3       x6 = k R3 sin z/R3
     x7 =   R4 cos t/R4       x8 =   R4 sin t/R4

Thus each point (x,y,z,t) in 4D spacetime represents two quaterions

              q1 = x1 + x3 + x5 + x7

              q2 = x2 + x4 + x6 + x8

To determine the absolute distances in this 8D space we again consider
the eight coordinate differentials, exemplified by

         d x1  =  i R1 (-sin(x/R1)) (1/R1) (dx)

(using the rule for total differentials) so the squared differentials
are exemplified by

           (d x1)^2  =  - sin^2(x/R1) (dx)^2

Adding up the eight squared differentials to give the square of the 
absolute differential interval leads again to the locally Lorentzian 
4D metric

       (ds)^2  =  (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2

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